see [1]}. Denote its image by \(\text{KHA}(Q,W)^{\prime }\) . It is a \(\mathbb {N}^I\times \mathbb {Z}\) -graded algebra and denote its \((d,w)\in \mathbb {N}^I\times \mathbb {Z}\) -graded part
by \(\text{KHA}(Q,W)^{\prime }_{d,w}\) .
For \(d\in \mathbb {N}^I\)  and \(w\in \mathbb {Z}\) ,
let \(P(d;\delta )_w\subset \text{KHA}(Q,W)^{\prime }_{d,w}\)  be the space of primitive-like elements with respect to \(\Delta \) , see Subsection REF  for a precise definition.
For \(A=(d_i, w_i)_{i=1}^k\)  in a set of partitions \(U^d_w(\delta )\) , define
\(P_A(\delta ):=\otimes _{i=1}^k P(d_i; \delta _{Ai})_{w_i},\) 
